[Spit]

Are you sure that the system is completely full? Some of your problems sound like you could have air in the system. Have you removed the radiator cap? and looked to see if coolant is up to the neck?

A good place to start would be to look at the coolant, does it appear to be in good condition? Is it rusty or any crud in it? You might start by draining out all the old coolant flushing the system real good. It's possible that there is "gunk" in the coolant that is blocking up your heater core and or some of the passages in the engine. On a Chevy Blazer my daughter owns I took the hoses off the heater core and back flushed it with a garden hose it opened it righ up an it heats like mad now. It didn't seem like there was that much gunk that came out but it made a big difference. relatively small particles can block off the flow through the heater core.

It could be a bad water pump but if it was low flow due to a bad pump, I would think that your engine would run hot all the time. Thermostat has nothing to do directly with the Heater, it opens when the engine temp gets high enough and then it send coolant through the radiator. Coolant is always running to the heater core. if your thermostat is stuck open the coolant will never get as warm as it should..but that doesn't explain the over-heating, with a thermo stuck open you should never overheat.

There are some vent doors in the ventilation system that open and close when changing from heat to cool..one of them could be stuck letting cold outside air in.

I would start with simple and flush and change the coolant.